title vibrational energy relaxation of azulene in the s-2 state. i. … · 2016. 6. 15. ·...

Title Vibrational energy relaxation of azulene in the S-2 state. I.Solvent species dependence

Author(s) Yamaguchi, T; Kimura, Y; Hirota, N

Citation JOURNAL OF CHEMICAL PHYSICS (2000), 113(7): 2772-2783

Issue Date 2000-08-15

URL http://hdl.handle.net/2433/49897

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Copyright 2000 American Institute of Physics. This article maybe downloaded for personal use only. Any other use requiresprior permission of the author and the American Institute ofPhysics.

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Kyoto University

Vibrational energy relaxation of azulene in the S2 state. I. Solventspecies dependence

T. Yamaguchi,a) Y. Kimura,b) and N. HirotaDepartment of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

~Received 24 January 2000; accepted 15 May 2000!

We have measured the time-resolved fluorescence spectra of azulene in theS2 state in compressedgases and in liquids. We have found that the band shape of the fluorescence changes significantlyin the earlier time scale after the photoexcitation when large excess energy~about 6500 cm21! isgiven. The change of the band shape is similar both in the compressed gases and in the liquids,although the time scales of the change are quite different. We have measured the excitation energydependence of the fluorescence band shape of the isolated molecule separately, and shown that thetime dependence of the fluorescence band shape in gases and liquids corresponds to the vibrationalenergy relaxation in theS2 state. Comparing with the excitation energy dependence of thefluorescence band shape of the isolated molecule, we have succeeded in determining the transientvibrational excess energy. The vibrational energy relaxation rates in theS2 state are 1–2 times fasterthan those in the ground state both in compressed gases and in liquids. ©2000 American Instituteof Physics.@S0021-9606~00!50231-1#

I. INTRODUCTION

Vibrational energy transfer to the bath molecules is oneof the fundamental reactions both in the gas phase1 and theliquid phase2,3 chemistry. A molecule can possess a largeamount of intramolecular vibrational excess energy just afterthe photoexcitation or the chemical reaction. Since vibra-tionally ‘‘hot’’ molecules show different reactivity, we needto know the rate of the energy dissipation in order to under-stand chemical reactions. Therefore, many experimental andtheoretical studies on the vibrational energy relaxation havebeen performed both in the gas phase and in solution.

In the gas phase, vibrational energy relaxation can betreated as a bimolecular reaction, and there have been twotopics on the vibrational energy relaxation in the gas phase.One is the relationship between the relative relaxation ratesand the molecular properties of both the solute and the bathmolecules,4–13 and the other is the dependence of the effi-ciency of the vibrational energy transfer on the vibrationalexcess energy.4–7,14–17

Although the same problems are also important in thevibrational energy relaxation in solution, the situation ap-pears more complicated in solution, since the vibrational en-ergy relaxation may no longer be the bimolecular reactiondue to the many-body effects in liquid. Many ideas on thevibrational energy relaxation in liquids have been proposedover the latter half of this century. The simplest idea amongthem is the isolated binary collision model~IBC model!,where the vibrational energy relaxation in solution is treatedas the bimolecular reaction between a solute and a solvent asin the gas phase reaction.2,3,5,18–29This model has been usedto analyze the experimental data since the earlier studies ofthe vibrational energy relaxation using the ultrasonic

experiment.2,22,23 After the first proposal, some criticismswere given to the IBC model,2,30,31 and many models havebeen proposed which include the many-body nature of theliquid, for example, three-body interference,31 hydrodynamicmodel,32 and heat diffusion model.3,33,34,35~c! In addition tothese theoretical models, many numerical simulation studieshave been performed in recent days owing to the great im-provement of the performance and the availability ofcomputers.19,27,29,36–39However, we consider that even thevalidity of the simplest and the earliest model has not beenmade clear to date. It is required to examine the IBC modeland clarify the relationship between the vibrational energyrelaxation in the gas phase and in the liquid phase.

In a series of papers, we shall present our experimentalresults on the vibrational energy relaxation of azulene~insetof Fig. 1! in the S2 state in gases, liquids, and so-calledsupercritical fluids. In paper I~this paper!, we introduce ourexperimental method and present the results in gases and inliquids. In paper II,40 we present the results in supercriticalfluids and discuss the validity of the IBC model. Parts of theresults of this work were already presented in the precedingletter,41 although some of the results are modified.

Although the vibrational energy relaxation in the groundelectronic state has been studied for various molecules, thenumber of the studies on the electronic excited state seemsrather small compared with that in the ground state in the gasphase. The vibrational energy relaxation in the electronic ex-cited state is more important in the solution chemistry. Sincethe vibrational energy relaxation in solution is very fast~within 100 ps!, only the fast photochemical reaction cancompete with it, where the electronic excited states are likelyto be involved. In addition, it is interesting to compare thevibrational energy relaxation in the excited states with that inthe ground state in order to clarify the relationship betweenthe relaxation rates and the properties of the solute mol-ecules. For this purpose, it is desirable to measure the relax-ation rates both in the ground and in the excited states of the

a!Present address: Institute for Chemical Research, Kyoto University,Gokasho, Uji, Kyoto 611-0011, Japan.

b!Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 7 15 AUGUST 2000

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samemolecules, because we can change the properties of thesolute keeping the structure of the molecule almost the same.However, even in the gas phase, there are only a few mol-ecules whose vibrational energy relaxation is measured bothin the ground and the excited states.7,12,42In solution, there isno case in which the relaxation from the vibrationally highlyexcited state is measured in both the ground and the elec-tronic excited states to our best knowledge. An electronicexcited state should relax to the ground state faster than thevibrational energy relaxation in order to create vibrationally‘‘hot’’ ground states by internal conversion. On the otherhand, the lifetime of an excited state should be longer thanthe vibrational energy relaxation in order to measure the vi-brational energy relaxation in the excited state. These twoconditions appear contradicting. However, azulene is a rarecandidate to satisfy both by using two different electronicexcited states so far as we know. The vibrational energyrelaxation of azulenein the ground stateis measured bymany researchers both in gases4,5,6,43 and in liquids.3,5,20,33

Recently, Schwarzeret al. measured the vibrational energyrelaxation of azulene in the ground state in supercriticalethane, carbon dioxide, and xenon.5 In this paper, these re-sults are compared with our results on the vibrational energyrelaxation in theS2 state.

Azulene emits relatively strong and long-lived~about 2ns! fluorescence from theS2 state, contrary to the fact thatthe internal conversion from higher electronic excited statesto theS1 state is usually very fast and that the fluorescencefrom higher states is very weak~Kasha’s Rule!. The absorp-tion and the fluorescence spectra of azulene are shown inFig. 2. Due to the peculiar properties of theS2 fluorescenceof azulene, there are many spectroscopic and theoreticalstudies on it.44–47For example, Hirata and Lim reported thatthe fluorescence band shape changes drastically with the ex-citation wavelength in the vapor phase.45 It means that thefluorescence band shape can be a measure for the intramo-lecular vibrational energy, and we can determine the timeevolution of the intramolecular vibrational energy from thetemporal change of the spectral band shapes. In this study,we measured the time-resolved fluorescence spectra of azu-lene in theS2 state in compressed gases and in liquids. Wefound temporal changes of the spectrum bandshape both incompressed gases and solutions. We also measured the exci-

tation energy dependence of the fluorescence bandshape ofan isolated molecule~vapor phase!, and found that the timeevolution of the spectrum band shape in solution correspondsto the loss of the intramolecular vibrational excess energy.By comparing the band shapes at different times with the‘‘hot’’ spectra of an isolated molecule, we can determine theabsolutevalue of the intramolecular vibrational excess en-ergy. The experimental scheme is drawn in Fig. 1. Althoughit has been frequently reported that there is a temporalchange of the fluorescence at 10 ps scale after the photoex-citation in solution, which has been assigned to the vibra-tional energy relaxation of the excited state,3,48,49,50this is thefirst study on the correspondence between thetransient‘‘hot’’ spectrum in solution and the ‘‘hot’’ fluorescence ofthe isolated molecule to our best knowledge. We shall notehere that the vibrational energy relaxation rates of azulene inthe S2 state in rare gas matrices were already reported.51

Hereafter in this paper, we will show the studies on the vi-brational energy relaxation of azulene in theS2 state in com-pressed gases and in liquids. The results in supercritical flu-ids will be presented in Paper II.40

II. EXPERIMENT

The apparatus to measure the time-resolved fluorescencespectrum by a streak camera is described elsewhere.41,52

Briefly, the second harmonics of a mode-locked Nd:YAGlaser~Coherent Antares 76! pumped a dye laser with a cavitydumper build in it~Coherent 700!. Saturable absorber~DAS-BTI! was used to stabilize the dye laser when the fast relax-ation in liquids was measured.53 The output pulse of the dyelaser was about 7 ps width~;3 ps width when saturableabsorber was used!, ;8 nJ, and 1.3 MHz of repetition whenoperated at 569 nm. After the polarization was adjusted by a

FIG. 1. Experimental scheme. IC, internal conversion; VR, vibrational en-ergy relaxation. The structure of azulene is drawn at the right-upper side.

FIG. 2. S0–S2 electronic spectra of azulene.~a! Absorption spectra;~b!fluorescence spectra. From upper to lower, in cyclohexane, in methanol, inwater, and in the vapor phase.

2773J. Chem. Phys., Vol. 113, No. 7, 15 August 2000 Vibrational energy relaxation of azulene. I


half wave plate and a polarizer, the output of the dye laserwas frequency doubled by a BBO crystal of 1 mm thickness.The obtained light of 285 nm was used for excitation. Thepolarization of the excitation light was adjusted to the magicangle so as to avoid the effect of rotational relaxation.54 Thefluorescence was collected at the right angle and focusedonto the slit of the spectrograph~Chromex 250IS!. A quartzdepolarizer was placed in front of the slit of the spectrographand both of the polarizations were collected. The width ofthe slit was 100mm ~optical slit width was about 3 nm! in allthe measurements.5 The spectra were obtained by a streakcamera~Hamamatsu, C4334! operated at the photon count-ing mode. A small amount of the SHG of the Nd:YAG laserwas separated to trigger the streak camera. The full-width athalf-maximum~FWHM! of the response function was typi-cally 30 ps when the streak camera was operated at the fast-est sweep speed. We had to reduce the sweep speed in orderto measure the slow vibrational energy relaxation in dilutegases, and the FWHM of the response function increases to100 ps in these experiments. The wavelength was calibratedby measuring the light from a low-pressure mercury lamp.55

The wavelength dependence of the sensitivity was correctedby measuring the reference sample~2-aminopyridine!.56 Thewavelength dependence of the time delay due to the me-chanical artifact of the streak camera was corrected by usingthe time profile of the fluorescence of azulene in the vaporphase. The dye laser was operated at 569 nm in the measure-ment of the time-resolved spectra in solution. The wave-length of the output of the dye laser was swept from 566 to676 nm in the measurement of the excitation wavelengthdependence in the vapor phase. Rhodamine 6G was used atthe wavelength shorter than 620 nm, and DCM was used atthe longer wavelength. Pyridine 1 was also used to operate at674 nm.

In order to measure the vibrational energy relaxation inthe S2 state, the relaxation rate should be faster than thelifetime of the S2 state. Therefore we compressed buffergases to increase the vibrational energy relaxation rates~0.4–10 MPa!. The high-pressure optical cell used for themeasurement in compressed gases is described elsewhere.57

The temperature of the cell was controlled at 34161 K byflowing thermostated water through the cell and measured bya thermocouple immersed in the cell. The pressure of the cellwas monitored by strain gages~Kyowa PGM 20 KH for,2MPa, and Kyowa PGM 500 KH for.2 MPa!, and the den-sities of the gases were calculated from the empirical equa-tions of state.58 The concentration of azulene was,1024 mol dm23. The sample cell was purged twice by thesample gas before experiment. Helium, argon, xenon, nitro-gen, ethane, and carbon dioxide were used as bath gases. Wealso measured the time-resolved fluorescence in oxygen inorder to estimate the electronic quenching rate by oxygen.An ordinary 10 mm quartz cell was used in the measurementof liquid solutions. Cyclohexane, acetonitrile, methanol, eth-ylene glycol, and water were used as solvents. In measuringthe fluorescence in the vapor phase of azulene, a smallamount of azulene was enclosed in the quartz cell that wasevacuated by a vacuum pump to 531023 Torr with azulenetrapped by liquid nitrogen. Then the cell was immersed into

an air bath that was temperature controlled at 34361 K. Inmeasuring the fluorescence in water vapor, a small amountof water was also enclosed in the vacuum quartz cell togetherwith a small amount of azulene. The cell was immersed intothe air bath and the temperature was controlled at 34361 or35361 K. The amount of the water was large enough thatthe pressure of water was calculated from the saturation va-por pressure. The decrease of the vapor pressure due to thesolute was neglected because of the low solubility of azuleneto water. The absorption spectra were measured by an UVabsorption spectrometer~UV2500PC, Shimadzu!.

Azulene ~Nacalai Tesque! was purified twice by subli-mation before use. Distilled water~Nacalai Tesque!, cyclo-hexane, acetonitrile, methanol~Nacalai Tesque, spectro-scopic grade!, and ethylene glycol ~Nacalai Tesque,guaranteed grade! were used as received. Helium~Iwatani,.99.999%!, argon ~Sumitomo Seika,.99.999%!, xenon~Iwatani, .99.995%!, nitrogen ~Sumitomo Seika,.99.999%!, ethane~Sumitomo Seika,.99%!, carbon diox-ide ~Sumitomo Seika,.99.98%!, and oxygen~Izumi, 99%!were used without further purification.

III. RESULTS

A. The excitation energy dependence of fluorescencein the vapor phase

Figure 3~a! shows typical examples of the fluorescencebandshapes at different excitation wavelengths in the vapor

FIG. 3. ~a! Excitation energy dependence of the fluorescence spectrum ofazulene in theS2 state. The excitation wavelengths are, from upper to lower,283 nm, 305 nm, 320 nm, and 338 nm, respectively. The lowest one is therelaxed fluorescence in 20 bar of Ar. Filled circles, experiment; solid lines,fitted ones.~b! The absorption spectrum of azulene in the vapor phase.Vertical lines indicate the energy of the photon used to measure the excita-tion energy dependence of the fluorescence line shapes in the vapor phase.The arrow indicates the photon energy used in the experiments in solution.

2774 J. Chem. Phys., Vol. 113, No. 7, 15 August 2000 Yamaguchi, Kimura, and Hirota


phase. We measured the fluorescence spectra at 283, 285,294, 305, 313, 320, 331, 337, and 338 nm. We did not detectmeaningful time dependence of the fluorescence bandshapes. The relaxed fluorescence in the dilute gas~argon, 20bar! is also shown. The bandshapes of the relaxed fluores-cence did not depend on buffer gases. The absorption spec-trum of azulene is shown in Fig. 3~b!, together with thephoton energy used to measure the excitation energy depen-dence. As is clearly shown, the fluorescence spectrum isbroad and structureless when excited at a shorter wavelength,i.e., the intramolecular vibrational excess energy is large.The fluorescence spectra become sharp and structured with adecrease of the intramolecular vibrational excess energy. Inaddition to the broadening, a small peak shift and the changeof the relative intensity of the vibronic bands are observed.These changes are consistent with the result reported byHirata and Lim.45

We also measured the excitation energy dependence ofthe fluorescence lifetime in the vapor phase. The results areshown in Fig. 4, together with the values determined byHirata and Lim, from the relative fluorescence quantumyield.45 The fluorescence lifetime gets shorter with an in-crease of the intramolecular vibrational excess energy, andthe excess energy dependence is almost exponential. Our ex-cess energy dependence is a little larger compared with thatof Hirata and Lim.45 We approximated the vibrational excessenergy (Eex) dependence of the fluorescence lifetime (tF)by the following function,

tF5tF0 exp~2aEex! ~1!

and obtained values aretF052.9 ns anda54.631024 cm.

B. Time-resolved fluorescence in gases and liquids

Figure 5 shows the temporal change of the spectrumband shape of azuleneS2 fluorescence in compressed argon~20 bar! and methanol at the ambient condition. Although thetime scales are quite different, the ways of the spectralchange resemble to each other. Similar changes are found inall the compressed gases and liquids, although the time scaleand the amount of the change depend on the species and thedensities of the solvents. Compared with the excitation en-ergy dependence of the fluorescence band shape in the vaporphase@Fig. 3~a!#, it is noticed that the fluorescence bandshapes at earlier times in solution have the characteristics of

the vibrationally unrelaxed spectra, i.e., broad, structureless,small red shift, and the change of the relative intensity of thevibronic bands. Therefore we can assign the change of thefluorescence bandshape to the vibrational energy relaxationof azulene in theS2 state.

In order to obtain the vibrational excess energy at eachtime delay, we compared the fluorescence line shapes withthose obtained in vapor phase with different vibrational ex-cess energy. In such a comparison, we have made a calibra-tion curve to convert the fluorescence band shape to the in-tramolecular excess energy; we fitted the spectra observed inthe vapor phase by an analytical function of the excess en-ergy whose derivatives we can obtain easily. First, the fluo-rescence spectra were normalized by the integrated intensity.Next, the fluorescence intensity at each probe wavelengthwas fitted by a third order polynomial of the excess energy.59

In this procedure, we used the relaxed fluorescence in replaceof the fluorescence excited at the 0–0 band, and the fluores-cence of zero excess energy was fixed. We used 348.2 nmfor the wavelength of the 0–0 band to calculate the intramo-lecular vibrational excess energy. The fitting curves are alsoshown in Fig. 3~a!. The experimental spectra are reproducedquite well by the fitting curves.

By using this calibration function, we fitted the fluores-cence spectrum at each delay time to determine the vibra-tional excess energy at each time. Since the relaxed fluores-cence spectrum of azulene depends on solvents@Fig. 2~b!#,we need to correct the spectrum change due to the solvent. Inthe correction, we took both the peak shift and the broaden-ing into account. First, we simply shifted the reference spec-tra ~spectra in the vapor phase!. Next, we convoluted thereference spectra with the following Gaussian function:

exp~2v2/4kBTls!/A2pkBTls, ~2!

FIG. 4. Intramolecular vibrational excess energy dependence of the fluores-cence lifetime in the vapor phase.d, this work;s, the lifetimes determinedfrom the relative fluorescence quantum yield by Hirata and Lim~Ref. 45!.

FIG. 5. Time-resolved fluorescence spectra of azulene.~a! In 20 bar ofargon. From upper to lower, at 105 ps~estimated excess energy, 5320cm21!, 306 ps~2890 cm21!, 506 ps~1460 cm21!, and 995 ps~490 cm21!. ~b!In methanol. From upper to lower, at 0.4 ps~2040 cm21!, 21.2 ps~520cm21!, and 41.9 ps~40 cm21!. Filled circles, observed; solid curves, fitted.



wherev, kB , T and ls stand for the frequency, the Boltz-mann constant, the absolute temperature, and the solvent re-organization energy, respectively. After the correction, wefitted the transient spectra to the reference spectra by theleast square method. The amounts of the shift and the broad-ening were determined so that the obtained vibrational ex-cess energy reaches to zero after the completion of the vibra-tional energy relaxation. We assumed that the shift and thebroadening do not depend on the vibrational excess energy.In compressed gases, no broadening was required and theamount of the shift was within6 1 nm. In the case of liquidsolution, larger reorganization energies were required for po-lar and protic solvents. For example,ls is about 70 cm21 formethanol and about 150 cm21 for water. The absolute valuesof the reorganization energies do not contradict with the sol-vent induced fluorescence Stokes shift~Fig. 2! qualitatively.The results of the fitting are shown in Fig. 5~solid curves!.The fitting procedure works well both in compressed gasesand in liquid solutions.

C. Vibrational energy relaxation in compressed gases

The time dependence of the intramolecular vibrationalenergy obtained by this fitting procedure is shown in Fig. 6.In the case of argon at 20 bar, the vibrational energy relax-ation time is about 500 ps as seen in Fig. 6. Since we knowthe excitation wavelength~285 nm! and the wavelength ofthe 0–0 band~348.2 nm!, we can calculate the intramolecu-lar vibrational excess energy just after the creation of theS2

state ~6430 cm21!. The initial vibrational energy obtainedfrom the time-resolved spectrum agrees with the theoreticalvalue well in the case of compressed gases. Therefore weconsider that the vibrational energy dissipation does not oc-cur in the time scale faster than the time resolution of ourexperiment in the case of compress gases. The time evolu-

tion of the vibrational excess energy is reproduced by asingle exponential function fairly well~solid curves in Fig.6!. The agreement is especially good for higher solvent den-sities. The deviations are found for lower solvent densityaround the vibrational excess energy of 5000 cm21 and 1000cm21. The spectrum quality is worse at lower densities dueto the lower solubility and the shorter excited-state lifetime.Since the vibrational excess energy dependence of the fluo-rescence band shape is relatively small around this excessenergy, we consider this deviation is due to the fitting errorof the fluorescence spectra. We interpret the time constant ofthis exponential function as the vibrational energy relaxationrates in compressed gases.

The vibrational energy relaxation rates determined bythe band shape analysis are plotted against the solvent den-sity in Fig. 7. The molecular gases work as efficient vibra-tional energy acceptors compared to rare gases. The com-parison of the relaxation efficiency in theS2 state with that inthe S0 state is given in Table I. Ideally, the biomolecularrates should be measured in the low-density limit. However,the relaxation rate is not proportional to the solvent densityin this density region, and the measurement at the lowerdensity is limited by the lifetime of theS2 state unfortu-nately. Therefore, we compare the relaxation rates ofS0 andS2 states at 1 mol dm23 instead. The rates in nitrogen arecompared at 2 mol dm23 due to the lack of the data of theS0

state. The relaxation rates of theS0 state are drawn from Ref.5. The Lennard-Jones collision frequency (ZLJ) is calculatedfrom the LJ parameters in Ref. 4~a!, assuming the propor-tionality between the density and the collision frequency.60

FIG. 6. Time dependence of the intramolecular vibrational excess energy indilute gases.~a! Results in argon. Filled circles, experiment at 20 bar; opencircles, that at 80 bar; solid curves, simulation.~b! Results in ethane. Filledcircles, experiment at 8 bar; open circles, that at 21 bar; solid curves, simu-lation.

FIG. 7. Solvent density dependence of the vibrational energy relaxationrates (kc) in dilute gases.d, helium;s, argon;j, xenon;h, nitrogen;l,ethane;L, carbon dioxide.

TABLE I. The efficiency of the vibrational energy transfer per collision.The relaxation rates in theS0 state are drawn from Ref. 5. The quenchingrate at 1 mol dm23 is used except for water vapor~vapor pressure! andnitrogen~2 mol dm23!. The collision frequency (ZLJ) is obtained from theLJ parameter in Ref. 4~a!.

Solvent gases kc(S0)/ZLJ(1023) kc(S2)/ZLJ(1023) kc(S2)/kc(S0)

Helium 4.9 4.7 1.0Argon ¯ 10 ¯

Xenon 5.5 12 2.2Nitrogen 6.9 14 2.0Ethane 19 30 1.6Carbon dioxide 13 28 2.2Water ¯ 43 ¯



Before the presentation of the results in liquids, we willdiscuss the time profile of the fluorescence intensity. In Fig.8, we show the time dependence of the total fluorescenceintensity in argon. The longitudinal axis is drawn in the logscale. A small dip near 1800 ps is an artifact due to thedamage of the MCP plate of our streak camera. As is clearlyseen, the fluorescence intensity does not decay with a singletime constant, but the decay rate becomes slower at latertimes. It is consistent with the fact that theS2 lifetime be-comes shorter with an increase of the intramolecular vibra-tional excess energy~Fig. 4!. The fluorescence decays fasterat earlier times because a large amount of the vibrationalexcess energy is stored in the solute molecule. The nonexpo-nential decay is found in all the gases, and the amplitudes ofthe fast components become smaller with an increase of thesolvent density, which is consistent with the above idea.7,61

In principle, we can determine the vibrational energydecay rate by the analysis of the fluorescence intensity, as isdemonstrated for the vibrational energy relaxation rates ofT1

pyrazine by McDowellet al., although in their case theyused the excess energy dependence of the intersystem cross-ing rate.7 Here, we tried to simulate the time dependence ofthe total fluorescence intensity. In the simulation, we firstassume exponential dependence of the fluorescence lifetimeon the intramolecular vibrational energy@Eq. ~1!#. Therefore,the time evolution of the fluorescence intensity (f (t)) obeysthe following equation:

d

dtf ~ t !52 f ~ t !/tF~e~ t !!, ~3!

where e(t) stands for the excess energy possessed by theexcited molecules, andtF is described by Eq.~1!. The timeevolution ofe(t) is assumed to be single exponential as fol-lows:

d

dte~ t !52kce~ t !, ~4!

where kc is the vibrational energy relaxation rate. In thesimulation, we usekc determined by the band shape analysis,and e(0) is fixed to 6430 cm21. The response function isassumed to be the following form:

g~ t !}sech2S t2t0

tRD , ~5!

where t0 is the peak of the response function, andtR is ameasure of the time resolution. The experimentally observedtotal fluorescence intensity (I F(t)) is expected to be de-scribed by the following equation:

I F~ t !5E2`

t

dt8g~ t8! f ~ t2t8!. ~6!

At first, we tried to reproduceI F(t) with the vibrational ex-cess energy dependence obtained from the experiment in thevapor phase~a54.631024 cm andtF

052.9 ns!. However,the simulation does not work at all~the solid curve in Fig. 8!.Even the fluorescence lifetime after the vibrational relaxation~1.5 ns in 20 bar of argon! disagrees.I F(t) could be simu-lated only by empirically adjustinga andtF

0 only to evaluatethe response function, and the results are shown in Fig. 8(a53.231024 cm andtF

051.5 ns!. The I F(t)’s for differentdensities and different gases are also reproduced by adjustinga53.2– 4.631024 cm and optimizingtF

0. Therefore, we uti-lized the fluorescence intensity only to estimate the systemresponse function. The origin of the discrepancy will be dis-cussed in Sec. IV A.

D. Vibrational energy relaxation in solution

We show in Fig. 9~a! the time profile of the intramolecu-lar vibrational energy of azulene in theS2 state in methanoldetermined from the spectrum band shape~filled circles!.The vibrational energy relaxation is almost completed withinthe instrumental response function. Therefore we cannot de-termine the relaxation rates simply by fitting the time profileof the excess energy to an exponential function, as is done inthe case of compressed gases. Hence we made a convolution

FIG. 8. Time dependence of the total fluorescence intensity in the 20 bar ofargon. Filled circles, experiment; solid curve, simulated by the excess en-ergy dependence of the fluorescence lifetime in the vapor phase; brokencurve, fitted by adjusting parameters~see text!.

FIG. 9. Time dependence of the intramolecular vibrational excess energy inmethanol.~a! Filled circles, experimental excess energy; solid curve, simu-lated excess energy, open circles, experimental fluorescence intensity; dot-ted lines, simulated fluorescence intensity; broken line, response function.~b! Simulation of^E(t)& with various relaxation time. Solid curves are thesimulated ones. The relaxation times are, from upper to lower, 8 ps, 7 ps, 6ps, 5 ps, and 4 ps, respectively. Filled circles, experiment; broken line,response function.



with the system response function as described below. First,we assumed that the vibrational excess energy obtained fromthe spectrum band shape corresponds to the average vibra-tional excess energy possessed by the excited molecules(Ê(t)&). Then Ê(t)& is expected be described by the fol-lowing equation:

I F~ t !Ê~ t !&5E2`

t

dt8g~ t8! f ~ t2t8!e~ t2t8!, ~7!

whereI F(t), g(t), f (t), ande(t) are defined in the previoussubsection.I F(t) is described by Eq.~6!. From Eqs.~6! and~7!, we obtained the relaxation ratekc . Equation~7! is notexact in the presence of the excess energy dependence of theexcited state lifetime. However, as will be shown in Appen-dix, this equation holds quantitatively well in our experimen-tal condition. In the actual fitting procedure, we guesskc atfirst, and optimizeI F(t) by Eqs.~3!–~6! in order to obtaint0 , tR , andtF

0. Next, we obtainkc from I F(t)Ê(t)& by Eqs.~3!, ~4!, and ~7!. Then, above two fitting procedures are re-peated until the self-consistent values are obtained. Thevalue ofa was fixed to 4.631024 cm. The change ofa didnot lead to a meaningful change of the relaxation rate, be-cause the internal conversion from the vibrationally ‘‘hot’’state is more than ten times slower than vibrational energyrelaxation in liquid. The initial value of the vibrational ex-cess energy was fixed to the same value as in compressedgases, 6430 cm21, neglecting the spectral shift in solution@about15 nm ~;2400 cm21! for fluorescence#. The fittingcurves and the response function are shown in Fig. 9~a!.Both I F(t) and Ê(t)& are reproduced fairly well by thisanalysis. In Fig. 9~b!, we simulatedÊ(t)& with differentvalues ofkc . The fitting error is estimated to be about61 ps.To be noted is that we could determine the relaxation ratebecause we evaluated theabsolute valuesof the intramolecu-lar vibrational excess energy, and assumed the single expo-nential decay function. Any one of the five curves in Fig.9~b! can reproduce the experiment fairly well if the absolutevalues of the excess energy are scaled. The relaxation ratesin the liquids are summarized in Table II.

E. Vibrational energy relaxation in water vapor

As is shown in Table II, the vibrational energy relaxationbecomes faster in protic solvents. The fast vibrational cool-ing in hydrogen bonding liquids is also found in othersystems.35,37,38,62,63On the other hand, it is widely knownthat the water molecule acts as an efficient vibrationalquencher in the gas phase.7,8 Therefore, we tried to measure

the vibrational energy relaxation of azulene in theS2 state inwater vapor in order to understand the reason for the fastenergy relaxation in liquid water.

The time-resolved spectra in the saturated water vapor at353 K are shown in Fig. 10~a!. The time profiles of thefluorescence intensity and the intramolecular vibrational ex-cess energies are shown in Fig. 10~b!. Due to the lower pres-sure of water vapor at 353 K~355 Torr!, the vibrationalenergy relaxation is slower than those in other compressedgases in this study. As a result, the population of theS2 statedisappears before the vibrational energy relaxation com-pletes, which makes the spectra att.1 ns noisy due to thesmall intensity. However, we consider that we can estimatethe relaxation rate roughly from this experiment, and we ap-proximateÊ(t)& to an exponential function@solid curve inFig. 10~b!#. The value of E(t)& is fixed to 6430 cm21. Thevibrational energy relaxation time is estimated to be 1.6 ns.We performed the same experiment at 343 K, where thesaturated vapor pressure is 234 Torr, and the vibrational en-ergy relaxation time 2.0 ns. By averaging these data, wecalculate the vibrational quenching efficiency per collision,which is shown in Table I. Water is an efficient vibrationalquencher also for azulene in theS2 state.

TABLE II. The vibrational energy relaxation times in liquids. The estimatederrors are about61 ps in the case of theS2 state.

Solvents kc(S0)21/psa kc(S2)21/ps kc(S2)/kc(S0)

Cyclohexane 13.7 11 1.3Acetonitrile 14.6 12 1.3Methanol 8.3 5.8 1.4Ethylene glycol 7.2 3.5 2.1Water ¯ 2.3 ¯

aReference 5.

FIG. 10. Vibrational energy relaxation in the saturated water vapor at 353K. ~a! Time-resolved spectra. From upper to lower, at 97 ps~estimatedexcess energy, 6070 cm21!, 499 ps~4560 cm21!, 1004 ps~3810 cm21!, and1504 ps~2250 cm21!, respectively. Marks, experiment; solid curves, fitting.~b! Time dependence of the vibrational excess energy. Filled circles, experi-ment, solid curve, fitting; broken curve, fluorescence intensity.



IV. DISCUSSION

A. The effect of other relaxation processes in theexcited states

In the analysis in Sec. III, we have implicitly made someassumptions on the relaxation processes in the excited state.First, the internal conversion from the higher singlet excitedstates to theS2 state is assumed to be very fast. The internalconversion from higher electronic excited states is generallyvery fast ~S2 azulene is a famous exception!, and the fluo-rescence from theS3 or theS4 state was not detected both inthe gas and the liquid phases.44 Therefore, we believe thatthe internal conversion to theS2 state is completed withinsub-ps.

The next assumption is the fast intramolecular vibra-tional redistribution~IVR! in theS2 state. We have evaluatedthe transient vibrational excess energy from a comparisonbetween the transient spectra and the time-integrated hotspectra in the vapor phase. However, since the fluorescencespectra are different before and after IVR,48 the transientspectra of the solute molecule is not characterized only bythe intramolecular vibrational excess energy. It is usuallyassumed in the vibrational cooling experiment that IVR com-pletes within sub-ps in solution. Ohtaet al. measured thetime-resolved fluorescence spectrum of Coumarin 481 byfluorescence up-conversion method, and found that IVR isalready completed just after the internal conversion from thehigher electronic excited state.48 Sarkaret al. found that theanisotropy of the fluorescence of tetracene in theS1 statedecays within 1 ps, which they ascribed to IVR in theS1

state.49 On the other hand, nonstatistical~IVR incomplete!vibrational population is sometimes observed by the time-resolved resonance Raman spectroscopy.64,65 However, fastIVR in the gas phase is reported in the case of azulene. In thejet-cooled experiment, Demmaret al. found that IVR of azu-lene in theS2 state is faster than 30 ps with a vibrationalexcess energy higher than 1950 cm21,66 and Diauet al. re-ported that the rate of IVR of azulene in theS2 state is 350 fswith an excess energy of 3800 cm21.67 In our experiment, wecould not find any significant change of the fluorescencespectra of the isolated azulene within the signal to noise ratioof our experiment. Therefore, we consider that IVR in theS2

state is fast enough that the state of the solute molecule ischaracterized well solely by the intramolecular vibrationalexcess energy at the vibrationally highly excited state. IVRmay cause some errors in the estimation of^E(t)& at smallvalues.

The third assumption is the neglect of the solvation dy-namics in polar solution. Although the solvent shift of theS2

spectra of azulene is small, the fluorescence Stokes shift isdetectable in polar solvents such as water~;3 nm!. There-fore, solvation dynamics such as dynamic Stokes shift mayaffect the observed transient spectra in polar solvents, whichwe did not take into account in the analysis of the transientspectra. However, the temporal changes of the fluorescencebandshape observed in polar solvents are not explainedsolely by the solvation dynamics. The solvation dynamicsshould appear as a red shift due to the energy dissipation. Itshould also appear as the broadening, not the narrowing, of

the fluorescence spectrum, since a narrow band laser selectsparticular solvation states. However, what we observed is ablue shift and a narrowing. In addition, the changes in therelative intensities of the vibronic bands are not likely ex-plained by the solvation dynamics. Therefore, the temporalchange of the fluorescence bandshape shown in Fig. 5~b! isdominated by the vibrational energy relaxation. However,there may be some contributions from the solvation dynam-ics in slowly relaxing solvents such as methanol or ethyleneglycol.68 The effect of solvation dynamics will act to de-crease the apparent excess energy and to increase the appar-ent relaxation rates. However, we neglect the effect of sol-vation dynamics simply because it is not distinguishableexperimentally from the vibrational cooling.

In connection with the solvation dynamics, solvent-induced thermochromism should be discussed as a candidatethat can change the fluorescence band shape. The increase ofthe local solvent temperature is expected after the heat re-lease to the solvent. The solvation structure under the el-evated temperature is different from that of the initial tem-perature, which can alter the fluorescence band shape. Forexample, Eq.~2! predicts an increase of the fluorescencebandwidth. However, we consider that this effect is small inmost cases of our study, since the absolute value of the sol-vent reorganization energy itself is small. The effect of thesolvent-induced thermochromism may be large in highly po-lar solvents as water. However, we did not take this effectinto account, because it is difficult to separate experimen-tally.

Finally, we should comment on the disagreement of thefluorescence lifetimes in the buffer gases and in the isolatedstate~Fig. 8!. We suspected that a small amount of oxygenthat remained in the cell quenched theS2 state of azulene. Inorder to test this idea, we measured the fluorescence lifetimein compressed oxygen~1–4 bar! and determined the bimo-lecular quenching rate. The rapid decrease of the lifetimewas found with an increase of the pressure of oxygen, andthe lifetime becomes 30 ps at 4 bar of oxygen. The fluores-cence band shape was the typical one for the vibrationallyunrelaxed molecule and did not change with time. Therefore,the electronic quenching occurs much faster than the vibra-tional quenching in compressed oxygen. The obtainedquenching rate of theS2 state is 5.6 ns21 bar21, which meansthat the quenching probability per collision is 0.5 if we cal-culate the collision frequency from the LJ parameter listed inRef. 4~a!. This value is close to the value reported for theS1

authracence~0.3!.69 Although the quenching rate may de-pend on the intramolecular vibrational energy, we use theabove rate as a measure of the rate for the vibrationally coolmolecule. If we ascribe the discrepancy of the relaxed fluo-rescence lifetime in Ar to that of vapor solely to the contami-nation of the oxygen, the partial pressure of oxygen is esti-mated to be 0.05 bar. Although we consider that the 0.05 barof oxygen is too large, the contamination of oxygen can beone of the reasons of the discrepancy of the relaxed lifetime.



In addition, the solvent effects on the lifetime may existsince our experimental condition is not so dilute.

B. The time profile of the vibrational energy relaxation

The experimental results ofÊ(t)& in compressed gases~Fig. 6! are almost reproduced by a single exponential func-tion in Eq.~4!, although there are small deviations. In the gasphase, the functional form ofÊ(t)& is related to the excessenergy~Ê&! dependence of the energy transfer efficiency percollision (^DE&), which is one of the major topics in the gasphase chemistry. The single exponential form ofÊ(t)&means that DE& is proportional toÊ&. The excess energydependence ofDE& has been measured for many molecules,and many polyatomic molecules show the proportionality be-tween Ê& and ^DE&.5,6 Although thresholds are found forsome molecules, they belong to rather special cases, sincethey are explained in terms of the coupling with upper elec-tronic states.7,15,17

Since so many molecules show the proportionality be-tweenÊ& and ^DE&, there must be a simple logic underly-ing. Here we show an explanation by the simplest model ofthe vibrational energy relaxation. Suppose that a harmonicoscillator is immersed in the heat bath~solvent molecules!.The perturbation from the heat bath on the oscillator is linearwith respect to the coordinate of the harmonic [email protected].,the form ofF(t)x, wherex is the coordinate of the oscillator,and F(t) is called force#. In this model, the vibrational en-ergy relaxation by only one phonon is allowed, and thedownward transition probability is proportional to the vibra-tional quantum number. It means the proportionality betweenÊ& and^DE&, since the number of phonon is proportional tothe vibrational energy. The above discussion holds irrespec-tive of the statistical properties ofF(t).70

We approximatedE(t)& as an exponential function alsoin liquids. It is nothing more than an approximation in thiscase, since we cannot determine the functional form ofÊ(t)& in liquids due to the low time resolution of our appa-ratus. The vibrational energy relaxation in liquids is some-times proposed to be multistep processes,3,33–35,64althoughno one observed the multi-step energy relaxation directly sofar as we know. Schwarzeret al. measured the vibrationalenergy relaxation ofS0 azulene in supercritical fluids, andreported that the decay of the hot band is a single exponentialat any density from gaslike to liquidlike ones.5 They reportedthe single exponential dependence also in liquids. For thesake of the comparison with their results, we employed thesingle exponential function asE(t)& in liquids.

C. The solute and the solvent dependence of theenergy relaxation rates in compressed gases

In this work, we found that the vibrational energyquenching of azulene in theS2 state is 1–2 times as efficientas that of azulene in the ground state in compressed gases.The fast energy relaxation in the electronic excited state hasalready been found in the case ofT1 pyrazine. Weismanet al. measured the vibrational energy relaxation ofT1 pyra-zine by the competitive radiationless decay method, andfound that the vibrational energy relaxation ofT1 pyrazine is

about seven times faster than that ofS0 pyrazine in the samegas.7 They proposed three reasons for the faster vibrationalenergy relaxation in theT1 state. The first one is the decreaseof the intramolecular vibrational frequency. Since the char-acteristic frequency of the classical solvent motion is muchlower than the frequency of the intramolecular vibration, thelower frequency mode is likely to relax faster. The secondone is the increase of the anharmonicity of the intramolecularvibration in the electronic excitation. The last one is the en-hancement of the coupling matrix elements due to the vi-bronic coupling to theT2 state, which they proposed is themajor reason for the fast relaxation ofT1 pyrazine, becausethey found a threshold near the energy level of theT2 state.We consider that the first two reasons apply also to the caseof azulene. Fujiiet al. reported the decrease of the intramo-lecular vibrational frequency in theS2 state.44 For example,n39 shifts from 326 cm21 to 236 cm21, and n17 does from402 cm21 to 372 cm21. Although there is a threshold neartheT2 level in the case ofT1 pyrazine, we could not find anytrace of the threshold near theS3 level in the case ofS2

azulene, and what we mainly observed is the vibrational en-ergy relaxation below theS3 level. Therefore, the third rea-son forT1 pyrazine by Weismanet al. does not apply to thevibrational energy relaxation of the present case. This maybe the reason for the different degree of the enhancement ofvibrational energy relaxation in the electronic excited state.

The solvent dependence of the quenching efficiency isanother problem. Roughly speaking, the tendency among thesolvent species is similar to those for other solute molecules.The molecular gases act as efficient quenchers comparedwith rare gases. The quenching efficiency of helium is lowerthan that of argon and xenon. Water acts as a very efficientquencher. However, when closely examined, the quenchingefficiency does not appear to be determined solely by theproperties of the solvent molecules, since the ratio of thequenching efficiencies in theS0 and S2 states varies withsolvent species. We do not have any further explanation ofthis difference at present. More experimental and theoreticalstudies will be required to clarify the meanings of this dif-ference.

We would like to comment here on the importance ofV–V transfer in the case of molecular quenchers. Jalenaket al. measured the vibrational energy relaxation ofS0 azu-lene in gaseous carbon dioxide by the time-resolved high-resolution IR spectroscopy.43 In their experiment, the vibra-tional hot bands, the rotational branches, and the Dopplerbroadening of thesolventmolecules were measured and theycould determine the relative importance of the acceptingmodes of solvents. They estimated that the ratio of theV–Vtransfer to the total energy transfer is about 25%. Althoughthe large vibrational quenching efficiency of molecular gasesis sometimes attributed to the existence of the intramolecularvibrational modes, their result indicates that theV–V trans-fer alone cannot explain the high quenching efficiency ofcarbon dioxide to azulene compared with those of rare gasmolecules. However, Fraelichet al. studied the vibrationalenergy relaxation ofS0 pyrazine in water vapor by a similarmethod, and concluded that the large quenching efficiency ofwater is due to the channels other than the rotational and the



translational modes.8 Anyway, we hope that this kind of ex-periments will clarify the mechanism of the vibrational en-ergy relaxation by the molecular gases.

D. The fast vibrational energy relaxation in thehydrogen bonding liquids

It is clearly seen from Table II that the vibrational en-ergy relaxation of azulene in theS2 state is enhanced in theprotic solvents. The fast vibrational energy relaxation in hy-drogen bonding solvents, especially in liquid water, is foundfor many other systems, and they are usually attributed to thehydrogen bonding. However, it is somewhat ambiguouswhether the solute–solvent or the solvent–solvent interaction~hydrogen bonding! is important. It is widely known that thecorrelation between water molecules due to the hydrogenbonding are strong enough that they have large influence onthe solvation of the solute molecule, for example, the posi-tive and the negative hydration, or the hydrophobic hydra-tion. Therefore, the hydrogen bonding between water mol-ecules can also affect the vibrational energy relaxation of asolute in liquid water. On the other hand, the vibrationalquenching efficiency of a water molecule per collision islarge in the gas phase. In addition, the solvent number den-sity of liquid water is larger than those of organic solvents.Therefore, the above two factors alone may explain the fastvibrational energy relaxation in liquid water.

In order to clarify the effect of the hydrogen bondingbetween solvents, we should estimate the relaxation rate inthe absence of the hydrogen bonding between the water mol-ecules. For this purpose, we tried to compare the vibrationalenergy relaxation in liquid water and in compressed helium.Since the LJ radius of water~0.271 nm! is similar to that ofhelium ~0.255 nm!,4~a! we employed a rough approximationthat the collision frequency between azulene and helium dif-fers from that between azulene and water only by the differ-ence of the reduced mass. Unfortunately, we do not have thevibrational energy relaxation rate of azulene in helium at thesame density as the liquid water~55 mol dm23!. If we ex-trapolate the linear density dependence of the vibrationalcooling rates ofS0 azulene in compressed helium in Ref. 5,the vibrational cooling time at 55 mol dm23 is estimated tobe 9 ps. Using this value, the IBC estimation of the coolingrate in liquid water is about 2 ps, which happens to agreewith the experiment. This indicates the importance of the gasphase quenching efficiency in the interpretation of the fastvibrational cooling in liquid water, although it does not nec-essarily exclude the importance of the hydrogen bonding be-tween solvent molecules, considering the rough estimationand the experimental uncertainty.

V. SUMMARY

In this work, we measured the time-resolved fluores-cence spectra of azulene in theS2 state in compressed gasesand liquids. The band shapes of the spectra change with timeat the earlier time after the photoexcitation both in gases andliquids, although the time scale of the change is quite differ-ent. From the comparison with the excitation energy depen-dence of the fluorescence band shape of the isolated mol-

ecule, we assigned the temporal evolution of the fluorescenceband shape to the vibrational energy relaxation. By analyzingthe temporal change of the fluorescence band shape, we de-termined the vibrational energy relaxation rates in theS2

state, which is 1–2 times as large as those in theS0 state. Weproposed that the low frequency shift of the intramolecularvibrational mode on the electronic excitation is a candidatefor the reason of the fast vibrational energy relaxation in theS2 state. In the liquid phase, we found that the vibrationalenergy relaxation in theS2 state is enhanced in the hydrogenbonding solvents. We also found that the water molecule isan efficient vibrational quencher for azulene in theS2 stateeven in the vapor phase. We showed that the large quenchingefficiency in the gas phase might be a reason for the fastenergy relaxation in liquid water. In order to clarify the re-lationship between the vibrational energy relaxation in thegas phase and that in the liquid phase, we will present thestudy on the vibrational energy relaxation in theS2 state ofazulene in so-called supercritical fluids in the successive Pa-per ~II !.40

ACKNOWLEDGMENTS

We are grateful to Dr. Y. Yoshimura~Kyoto University!for the use of the UV absorption spectrometer. T. Y. grate-fully acknowledges a research fellowship from the Japan So-ciety for the Promotion of Science~JSPS! for Young Scien-tists. This work is supported by CREST~Core Research forEvolutional Science and Technology! of Japan Science andTechnology~JST! and by the Research Grant-in-Aid fromthe Ministry of Education, Science and Culture~No.11640504!.

APPENDIX

As was mentioned in the text, Eq.~7! is not an exactrelationship, but a mere approximation under the presence ofthe excess energy dependent fluorescence lifetime. Supposethat there is a distribution of the vibrational excess energy inan ensemble of the excited molecules. Since the higher en-ergy part of the distribution decays faster due to the smallerlifetime, the averaged excess energy decreases with timeeven in the presence of no buffer gas. Therefore, the excessenergy dependence of the fluorescence lifetime makes theapparent relaxation rate faster. This effect becomes largerwhen the vibrational energy relaxation is slow, since theelectronic relaxation can compete with the vibrational energyrelaxation in such a case. In order to estimate this effectquantitatively, we have made a Monte Carlo simulation.

In this simulation, a solute molecule is subject to uncor-related collisions with buffer gas molecules. The distributionof the energy loss (P(E,DE)) at a collision is assumed to beexponential as follows:

P~E,DE!5H exp~2DE/^DE&~E!!/^DE&~E! ~DE.0!

0 ~DE,0!.~A1!



We consider that this exponential distribution suitable for thefirst approximation.6 The averaged energy loss (^DE&(E)) isproportional to the excess energy~E! in our model as fol-lows:

^DE&~E!5E

Ztc, ~A2!

wheretc andZ stand for the cooling time and the collisionfrequency, respectively. Upward collisions~collisional acti-vation! are not considered in our model for simplicity. In oursimulation, a solute molecule is prepared as the excited statewith an initial energy, 6430 cm21. Then collisions withbuffer gases are generated randomly with collision frequencyZ, and the vibrational excess energy (e(t)) decreases at eachcollision according to the probability distribution as Eq.~A1!. At the same time, the electronic relaxation from theS2

state occurs according to the energy dependent lifetime asEq. ~1! with experimentally determined values oftF

0 anda.Therefore, the survival probability (p(t)) of the S2 state de-creases with time. The averaged vibrational excess energy(Ê(t)&) in our experiment is expressed byp(t) ande(t) asfollows:

Ê~ t !&5^p~ t !e~ t !&

^p~ t !&. ~A3!

On the other hand, what we should obtain actually isê(t)&,which obeys the following time dependence:

ê~ t !&5e~0!expS 2t

tcD . ~A4!

These two functions do not agree with each other exactly, solong as the electronic relaxation and the vibrational excessenergy are correlated.

We have calculatedE(t)& with different values oftc ,and the results are compared with Eq.~A4! numerically. Thevalue ofZtc is fixed to 100, which is a typical value of ourexperiment~Table I!. The averages are taken for 1000 trajec-tories. The results of the simulations are shown in Fig. 11.The values of E(t)& agree well with Eq.~A4!. In particular,both functions are almost indistinguishable when the cooling

time (tc) is smaller than 1 ns. It indicates that the approxi-mation of Eq.~7! works quantitatively in most of our experi-mental conditions.

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FIG. 11. A Monte Carlo simulation of the apparent relaxation of the vibra-tional excess energy in the presence of the excited state lifetime that de-pends on the vibrational excess energy. Solid curve with filled circles standsfor the correct time dependence@Ê(0)&exp(2t/tc), where tc means thevibrational cooling time#. Dotted, dashed, dashed–dotted, and long-dashedcurves are the results of simulations attc5100 ps, 500 ps, 1 ns, and 2 ns,respectively.



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59In the previous letter~Ref. 41! we used the second order polynomialinstead. The fitting is not so good at the lower excess energy in the case ofthe second order polynomial, and this discrepancy leads to the overesti-mate of the relaxation rate in the high-density fluid.

60We consider that this assumption is not valid, and the invalidity appears asthe nonproportionality of the relaxation rates with the solvent density. Thenonproportionality has also been found for the vibrational energy relax-ation rates in the ground state~Ref. 5! and we consider that it is due to thenonlinear dependence of the collision frequency on the solvent density.Since the equations of state evidently deviates from that of ideal gas inthis density region, it is quite natural that the collision frequency is notproportional to the solvent density, either. However, we consider that thiserror in the estimation ofZLJ is small enough to discuss the relative ratesamong different quenchers and the difference between two electronicstates.

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69J. E. Haebig, J. Phys. Chem.71, 4203~1967!.70However, it should be noted that Rectoret al. reported that the population

decay time (T1) of overtone is smaller than the half ofT1 of fundamentalat the low temperature@K. D. Rector, A. S. Kwok, C. Ferrante, A. Tok-makoff, C. W. Rella, and M. D. Feyer, J. Chem. Phys.106, 10027~1997!#,althoughT1 in their experiment may not correspond to the intramolecularenergy transfer.



title vibrational energy relaxation of azulene in the s-2 state. i. … · 2016. 6. 15. ·...

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